X-ray scattering measurements of polycrystalline or amorphous samples may be achieved by transmitting X-rays from a source to the sample and detecting the diffracted, scattered X-rays. The measured X-ray scattering pattern from the polycrystalline or amorphous substance normally consists of the superposition of overlapping multiple peaks, or components. For crystalline substances the true, so-called angular Bragg position of the peak is a function of crystallographic unit cell. Individual components are subjected to influence of instrumental and spectral aberrations, deforming and shifting the component from the original (Bragg) position. The aberrations are generally different for different F components.
Analysis of patterns with strongly overlapping components deformed by the aberrations may be complicated or even impossible to perform correctly.
A further problem is that the comparison of data measured with different optics on different systems or even using the same system at different times can be difficult due to the difference in instrumental aberrations. This makes it difficult to compare data taken at different times or on different equipment.
In general, therefore, the dependence of measured data on aberrations reduces accuracy and reliability of analytical results.
One example is the measurement of samples with peaks at low angles, which is typical of measurements carried out on pharmacological or nano-scale materials. As the result of the influence of instrumental aberrations, these peaks are strongly deformed and shifted from their theoretical Bragg positions. The pharmaceutical substances often exhibit polymorphism, and as a result that the real Bragg positions of the peaks are slightly different for different polymorphic phases. The effect of instrumental aberrations creates serious obstacles for undertaking analytical tasks such as indexing the crystallographic unit cell, phase identification by searching and matching in the reference patterns or discriminating between polymorphs.
Correction for instrumental aberrations is therefore required to obtain physically consistent information from different components in the pattern.
There are a few different approaches to use the information on instrumental aberrations for the correction analysis.
One approach is based on the introduction of a model to simulate superposition of separate peaks determined by a crystalline unit cell model. Profiles of separate peaks are presented empirically as for example in the so-called “Rietveld” method.
This approach assumes that knowledge of crystal structure, the structure of the unit crystalline cell, and/or the atomic structure is available to obtain the model of the scattering pattern. The model than may be compared with the measured pattern and the best fit is obtained by varying parameters of the model. There are other methods based on the peak model for example the “LeBail” method, where the peak positions are dependent on the unit cell but the peak shapes and weights are independent.
The non-empirical method to calculate the model of the peaks uses superposition from “first principles” by mathematical simulation of the instrument as set out in V. A. Kogan and M. Kupryanov, J. Appl. Cryst (1992), 25, 16-25 “X-Ray Diffraction Line Profiles by Fourier Synthesis” which describes a way in which line profiles can be calculated in Fourier space.
An alternative approach devoted to the non-empirical simulation of peak shapes in real space was described by R. W. Cheary and A. Coelho, J. Appl. Cryst (1992), 25, 109-121. “A Fundamental Parameters Approach to X-ray Line-Profile Fitting”.
All these methods generally deal with a model of the diffraction peaks. While being powerful analytical tools, these approaches are not always convenient in the sense of the required input to the model.
Such calculations may be complicated and require a lot of operator input to obtain analytical result. For new materials with low symmetry, the complexity of the measured pattern may create obstacles for fast analysis or comparison.
There is a need therefore for a method to correct for aberrations without modelling the sample or the superposition of peaks, dealing with the pattern as a single continuum.
There are methods known in the art devoted to removing of Cu Kα2 and other spectral components of known shape. These methods generally based on identification of doublet or multiplet areas in the pattern and applying deconvolution in direct or Fourier space. These are also model based methods.
T. Ida and H. Toraya, J. Appl. Cryst, 35, 58-68 “Deconvolution of the instrumental functions in powder X-Ray diffractometry” and T. Ida, Rigaku Journal Volume 20, N2, December 2003 describe what appears to be a model independent approach for aberration correction. The authors suggest that they can correct the instrumental aberrations in the complete pattern. They use “rescaling” where for deconvolution of any instrumental function they are trying to find the scale transformation for 2θ angle such that this instrumental function would become constant. This approach does not require detailed knowledge of the crystal structure.
Unfortunately this approach is not general. It is of use to correct simple instrumental functions allowing correct rescaling to get the aberration constant.
Even though the method requires sophisticated analytical calculations and different scale conversions for the instrumental function applied, it is hardly applicable to anything other than very simple instrumental functions, and is accordingly unsuitable for most practically important functions of axial divergence, for correcting of any kinds of interacting functions such as functions of equatorial divergence interacting with sample transparency function for the cylindrical samples, etc.
It might be thought that an alternative useful way of reducing the aberrations would be to use appropriate optics. However, this would not be useful because it would imply the restriction of beam trajectories and therefore increase measurement time and decrease measurement quality.